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In many biological situations, a species arriving from a remote source diffuses in a domain confined between two parallel surfaces until it finds a binding partner. Since such a geometric shape falls in between two- and three-dimensional settings, the behavior of the macroscopic reaction rate and its dependence on geometric parameters are not yet understood. Modeling the geometric setup by a capped cylinder with a concentric disk-like reactive region on one of the lateral surfaces, we provide an exact semi-analytical solution of the steady-state diffusion equation and compute the diffusive flux onto the reactive region. We explore the dependence of the macroscopic reaction rate on the geometric parameters and derive asymptotic results in several limits. Using the self-consistent approximation, we also obtain a simple fully explicit formula for the reaction rate that exhibits a transition from two-dimensional to three-dimensional behavior as the separation distance between lateral surfaces increases. Biological implications of these results are discussed.
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We investigate the influence of a stochastically fluctuating step-barrier potential on bimolecular reaction rates by exact analytical theory and stochastic simulations. We demonstrate that the system exhibits a new resonant reaction behavior with rate enhancement if an appropriately defined fluctuation decay length is of the order of the system size. Importantly, we find that in the proximity of resonance the standard reciprocal additivity law for diffusion and surface reaction rates is violated due to the dynamical coupling of multiple kinetic processes. Together, these findings may have important repercussions on the correct interpretation of various kinetic reaction problems in complex systems, as, e.g., in biomolecular association or catalysis.
A self-consistent equation to derive a discreteness-induced stochastic steady state is presented for reaction-diffusion systems. For this formalism, we use the so-called Kuramoto length, a typical distance over which a molecule diffuses in its lifetime, as was originally introduced to determine if local fluctuations influence globally the whole system. We show that this Kuramoto length is also relevant to determine whether the discreteness of molecules is significant or not. If the number of molecules of a certain species within the Kuramoto length is small and discrete, localization of some other chemicals is brought about, which can accelerate certain reactions. When this acceleration influences the concentration of the original molecule species, it is shown that a novel, stochastic steady state is induced that does not appear in the continuum limit. A theory to obtain and characterize this state is introduced, based on the self-consistent equation for chemical concentrations. This stochastic steady state is confirmed by numerical simulations on a certain reaction model, which agrees well with the theoretical estimation. Formation and coexistence of domains with different stochastic states are also reported, which is maintained by the discreteness. Relevance of our result to intracellular reactions is briefly discussed.
Purpose: Diffusion-weighted steady-state free precession (DW-SSFP) is shown to provide a means to probe non-Gaussian diffusion through manipulation of the flip angle. A framework is presented to define an effective b-value in DW-SSFP. Theory: The DW-SSFP signal is a summation of coherence pathways with different b-values. The relative contribution of each pathway is dictated by the flip angle. This leads to an apparent diffusion coefficient (ADC) estimate that depends on the flip angle in non-Gaussian diffusion regimes. By acquiring DW-SSFP data at multiple flip angles and modelling the variation in ADC for a given form of non-Gaussianity, the ADC can be estimated at a well-defined effective b-value. Methods: A gamma distribution is used to model non-Gaussian diffusion, embedded in the Buxton signal model for DW-SSFP. Monte-Carlo simulations of non-Gaussian diffusion in DW-SSFP and diffusion-weighted spin-echo (DW-SE) sequences are used to verify the proposed framework. Dependence of ADC on flip angle in DW-SSFP is verified with experimental measurements in a whole, human post-mortem brain. Results: Monte-Carlo simulations reveal excellent agreement between ADCs estimated with DW-SE and the proposed framework. Experimental ADC estimates vary as a function of flip angle over the corpus callosum of the postmortem brain, estimating the mean and standard deviation of the gamma distribution as $1.50cdot 10^{-4} mm^2/s$ and $2.10cdot 10^{-4} mm^2/s$. Conclusion: DW-SSFP can be used to investigate non-Gaussian diffusion by varying the flip angle. By fitting a model of non-Gaussian diffusion, the ADC in DW-SSFP can be estimated at an effective b-value, comparable to more conventional diffusion sequences.
Many experiments in recent years have reported that, when exposed to their corresponding substrate, catalytic enzymes undergo enhanced diffusion as well as chemotaxis (biased motion in the direction of a substrate gradient). Among other possible mechanisms, in a number of recent works we have explored several passive mechanisms for enhanced diffusion and chemotaxis, in the sense that they require only binding and unbinding of the enzyme to the substrate rather than the catalytic reaction itself. These mechanisms rely on conformational changes of the enzyme due to binding, as well as on phoresis due to non-contact interactions between enzyme and substrate. Here, after reviewing and generalizing our previous findings, we extend them in two different ways. In the case of enhanced diffusion, we show that an exact result for the long-time diffusion coefficient of the enzyme can be obtained using generalized Taylor dispersion theory, which results in much simpler and transparent analytical expressions for the diffusion enhancement. In the case of chemotaxis, we show that the competition between phoresis and binding-induced changes in diffusion results in non-trivial steady state distributions for the enzyme, which can either accumulate in or be depleted from regions with a specific substrate concentration.
The reduced 1D Poisson-Nernst-Planck (PNP) model of artificial nanopores in the presence of a permanent charge on the channel wall is studied. More specifically, we consider the limit where the channel length exceed much the Debye screening length and channels charge is sufficiently small. Ion transport is described by the nonequillibrium steady-state solution of the PNP system within a singular perturbation treatment. The quantities, 1/lambda -- the ratio of the Debye length to a characteristic length scale and epsilon -- the scaled intrinsic charge density, serve as the singular and the regular perturbation parameters, respectively. The role of the boundary conditions is discussed. A comparison between numerics and the analytical results of the singular perturbation theory is presented.
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